Examining this list and supposing we have no knowledge of the possible contents of r5c9 we see that the four cells r1c9, r2c9, r6c9, and r9c9 (the "naked quadruple[t]") must contain the four values 3, 4, 6, and 9 in some order. Since the values 1, 7, and 8 are already filled in, there are only two possibilities left for r5c9 -- {2, 5}.

Another way to reason about this is to observe that {3, 4, 6, 9} must fit in r1c9/r2c9/r6c9/r9c9 in some order -- that's four values in four boxes. And from the list above, the "9" has to go at r6c9, leaving {3, 4, 6} in r1c9/r2c9/r9c9. dcb

Examining this list and supposing we have no knowledge of the possible contents of r5c9 we see that the four cells r1c9, r2c9, r6c9, and r9c9 (the "naked quadruple[t]") must contain the four values 3, 4, 6, and 9 in some order. Since the values 1, 7, and 8 are already filled in, there are only two possibilities left for r5c9 -- {2, 5}.

Another way to reason about this is to observe that {3, 4, 6, 9} must fit in r1c9/r2c9/r6c9/r9c9 in some order -- that's four values in four boxes. And from the list above, the "9" has to go at r6c9, leaving {3, 4, 6} in r1c9/r2c9/r9c9. dcb

Ehm, I think FROM the fact that r3c8 and r3c9 can only be {2,5} FOLLOWS that {3, 4, 6, 9} must fit in r1c9/r2c9/r6c9/r9c9 in some order. Not vice versa.